Questions: Larmor Formula for Radiated Power

Question 1

According to the Larmor formula, what happens to the radiated power if the charge's acceleration is doubled while everything else stays the same?

• A. The power doubles
• B. The power increases by a factor of 4
• C. The power increases by a factor of 8
• D. The power stays the same — only the frequency of radiation changes

Answer: (B)

The Larmor formula is P = q²a²/(6πε₀c³) — power is proportional to acceleration squared. Doubling a gives a factor of 2² = 4 increase in power. This quadratic dependence on acceleration is not arbitrary: the radiation field itself is proportional to a, so the energy flux (Poynting vector ∝ E²) is proportional to a², and integrating over a sphere gives the same a² dependence.

Question 2

In which direction does an accelerating point charge radiate the most power?

• A. Along the direction of acceleration — maximum emission in the direction of motion
• B. Perpendicular to the acceleration — the radiation pattern peaks at 90° from the acceleration axis
• C. Uniformly in all directions — the Larmor formula gives total power, not directional
• D. Opposite to the direction of acceleration — the charge pushes radiation away from itself

Answer: (B)

The angular distribution of Larmor radiation goes as sin²θ, where θ is measured from the acceleration axis. This means zero power is radiated along the acceleration direction (θ = 0° or 180°) and maximum power is radiated perpendicular to it (θ = 90°). The pattern is shaped like a donut with the acceleration axis as the central hole — this is the characteristic dipole radiation pattern.

Question 3

A charged particle moving at constant velocity in a straight line radiates electromagnetic energy according to the Larmor formula.

• T. True
• F. False

Answer: False

The Larmor formula is P = q²a²/(6πε₀c³) — power depends on acceleration squared. A charge at constant velocity has zero acceleration, so P = 0: it emits no radiation. Radiation requires acceleration. This is a crucial distinction: a moving charge creates electric and magnetic fields that propagate with it (the 'velocity fields' or 'Coulomb fields'), but these fields do not carry energy away to infinity — only the radiation fields, which require acceleration, do that.

Question 4

The Larmor formula predicts that an electron in a classical circular orbit around a nucleus should continuously lose energy, implying atoms cannot be stable in classical electrodynamics.

• T. True
• F. False

Answer: True

A circular orbit requires centripetal acceleration directed inward. The Larmor formula gives P = q²a²/(6πε₀c³) > 0 for any nonzero centripetal acceleration. This means the electron continuously radiates energy, losing it to electromagnetic radiation. As energy decreases, the orbital radius decreases and the centripetal acceleration increases, accelerating the radiation and causing a runaway spiral inward. Classical electrodynamics predicts atoms collapse in nanoseconds — this 'classical catastrophe' was a key motivation for quantum mechanics.

Question 5

Why do circular particle accelerators face a fundamental energy-loss challenge from the Larmor formula that linear accelerators avoid?

Model answer: In a circular accelerator, particles move in a curved path, which requires centripetal acceleration directed toward the center of curvature. Even if particle speed is constant, centripetal acceleration is nonzero, so the Larmor formula gives P > 0 — the particles continuously radiate (synchrotron radiation) and lose energy. A linear accelerator accelerates particles along a straight line, so the acceleration is parallel to velocity and there is no centripetal component persisting throughout the path; once the accelerating section ends, acceleration stops and radiation ceases.

Synchrotron radiation scales as a⁴ in the relativistic generalization of the Larmor formula, making it an increasingly severe problem at high energies. This is why high-energy electron accelerators (like LEP at CERN) eventually hit a wall where all the energy pumped in goes straight out as synchrotron radiation. The Large Hadron Collider uses protons instead of electrons precisely because protons are ~1800× heavier, giving far less centripetal acceleration for the same momentum, and hence far less synchrotron radiation.